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Infinite–Dimensional Cerebellar Controller

for Realistic Human Biodynamics

Vladimir G. Ivancevic∗ Tijana T. Ivancevic†

Abstract

In this paper we propose an ∞−dimensional cerebellar model of neural controllerfor realistic human biodynamics. The model is developed using Feynman’s action–amplitude (partition function) formalism. The cerebellum controller is acting as a su-pervisor for an autogenetic servo control of human musculo–skeletal dynamics, whichis presented in (dissipative, driven) Hamiltonian form. The ∞−dimensional cerebellarcontroller is closely related to entropic motor control.

Keywords: realistic human biodynamics, cerebellummotion control,∞−dimensionalneural network

Contents

1 Introduction 2

2 Sub-Cerebellar Biodynamics and Its Spinal Reflex Servo–Control 4

2.1 Local Muscle–Joint Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Hamiltonian Biodynamics and Its Reflex Servo–Control . . . . . . . . . . . 6

3 Cerebellum: The Adaptive Path–Integral Comparator 9

3.1 Cerebellum as a Neural Controller . . . . . . . . . . . . . . . . . . . . . . . 93.2 Hamiltonian Action and Neural Path Integral . . . . . . . . . . . . . . . . . 113.3 Entropy and Motor Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

∗Human Systems Integration, Land Operations Division, Defence Science & Technology Organisation,P.O. Box 1500, Edinburgh SA 5111, Australia (Vladimir.Ivancevic@dsto.defence.gov.au)

†School of Electrical and Information Engineering, University of South Australia, Mawson Lakes, S.A.5095, Australia (Tijana.Ivancevic@unisa.edu.au)

1

4 Appendix 14

4.1 Houk’s Autogenetic Motor Servo . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Cerebellum and Muscular Synergy . . . . . . . . . . . . . . . . . . . . . . . 164.3 Feynman’s Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1 Introduction

Realistic human biodynamics (RHB) is a science of human (and humanoid robot) motionin its full complexity. It is governed by both Newtonian dynamics and biological controllaws (see [Iva04, IB05, II06a, II06b, II06c]).

There are over 200 bones in the human skeleton driven by about 640 muscular ac-tuators (see, e.g., [Mar98]). While the muscles generate driving torques in the movingjoints,1 subcortical neural system performs both local and global (loco)motion control:first reflexly controlling contractions of individual muscles, and then orchestrating all themuscles into synergetic actions in order to produce efficient movements. While the localreflex control of individual muscles is performed on the spinal control level, the global inte-gration of all the muscles into coordinated movements is performed within the cerebellum

[II06a, II06b].All hierarchical subcortical neuro–muscular physiology, from the bottom level of a

single muscle fiber, to the top level of cerebellar muscular synergy, acts as a temporal

〈out|in〉 reaction, in such a way that the higher level acts as a command/control space forthe lower level, itself representing an abstract image of the lower one:

1. At the muscular level, we have excitation–contraction dynamics [Hat77a, Hat78,Hat77b], in which 〈out|in〉 is given by the following sequence of nonlinear diffusionprocesses [II06a, II06b]:

neural action potential synaptic potential muscular action potential

excitation contraction coupling muscle tension generating.

1Here we need to emphasize that human joints are significantly more flexible than humanoid robot joints.Namely, each humanoid joint consists of a pair of coupled segments with only Eulerian rotational degreesof freedom. On the other hand, in each human synovial joint, besides gross Eulerian rotational movements(roll, pitch and yaw), we also have some hidden and restricted translations along (X,Y, Z)−axes. Forexample, in the knee joint, patella (knee cap) moves for about 7–10 cm from maximal extension to maximalflexion). It is well–known that even greater are translational amplitudes in the shoulder joint. In otherwords, within the realm of rigid body mechanics, a segment of a human arm or leg is not properlyrepresented as a rigid body fixed at a certain point, but rather as a rigid body hanging on rope–likeligaments. More generally, the whole skeleton mechanically represents a system of flexibly coupled rigidbodies. This implies the more complex kinematics, dynamics and control then in the case of humanoidrobots.

2

Its purpose is the generation of muscular forces, to be transferred into driving torqueswithin the joint anatomical geometry.

2. At the spinal level, 〈out|in〉 is given by autogenetic–reflex stimulus–response control

[Hou79]. Here we have a neural image of all individual muscles. The main purposeof the spinal control level is to give both positive and negative feedbacks to sta-bilize generated muscular forces within the ‘homeostatic’ (or, more appropriately,‘homeokinetic’) limits. The individual muscular actions are combined into flexor–extensor (or agonist–antagonist) pairs, mutually controlling each other. This is themechanism of reciprocal innervation of agonists and inhibition of antagonists. It hasa purely mechanical purpose to form the so–called equivalent muscular actuators

(EMAs), which would generate driving torques Ti(t) for all movable joints.

3. At the cerebellar level, 〈out|in〉 is given by sensory–motor integration [HBB96]. Herewe have an abstracted image of all autogenetic reflexes. The main purpose of thecerebellar control level is integration and fine tuning of the action of all active EMAsinto a synchronized movement, by supervising the individual autogenetic reflex cir-cuits. At the same time, to be able to perform in new and unknown conditions, thecerebellum is continuously adapting its own neural circuitry by unsupervised (self–organizing) learning. Its action is subconscious and automatic, both in humans andin animals.

Naturally, we can ask the question: Can we assign a single 〈out|in〉 measure to allthese neuro–muscular stimulus–response reactions? We think that we can do it; so inthis Letter, we propose the concept of adaptive sensory–motor transition amplitude as aunique measure for this temporal 〈out|in〉 relation. Conceptually, this 〈out|in〉−amplitude

can be formulated as the ‘neural path integral ’:

〈out|in〉 ≡ 〈motor|sensory〉amplitude

=

∫

D[w, x] ei S[x]. (1)

Here, the integral is taken over all activated (or, ‘fired’) neural pathways xi = xi(t) of thecerebellum, connecting its input sensory−state with its outputmotor−state, symbolicallydescribed by adaptive neural measure D[w, x], defined by the weighted product (of discretetime steps)

D[w, x] = limn→∞

n∏

t=1

wi(t) dxi(t), (2)

in which the synaptic weights wi = wi(t), included in all active neural pathways xi = xi(t),are updated by the standard learning rule

new value(t+ 1) = old value(t) + innovation(t).

3

More precisely, the weights wi in (2) are updated according to one of the two standardneural learning schemes, in which the micro–time level is traversed in discrete steps, i.e.,if t = t0, t1, ..., tn then t+ 1 = t1, t2, ..., tn+1:

2

1. A self–organized, unsupervised (e.g., Hebbian–like [Heb49]) learning rule:

wi(t+ 1) = wi(t) +σ

η(wd

i (t)− wai (t)), (3)

where σ = σ(t), η = η(t) denote signal and noise, respectively, while superscripts dand a denote desired and achieved micro–states, respectively; or

2. A certain form of a supervised gradient descent learning :

wi(t+ 1) = wi(t)− η∇J(t), (4)

where η is a small constant, called the step size, or the learning rate, and ∇J(n)denotes the gradient of the ‘performance hyper–surface’ at the t−th iteration.

Theoretically, equations (1–4) define an∞−dimensional neural network (see [IA07, IAY08,II08c]). Practically, in a computer simulation we can use 107 ≤ n ≤ 108, roughly corre-sponding to the number of neurons in the cerebellum [II07a, II07b].

The exponent term S[x] in equation (1) represents the autogenetic–reflex action, de-scribing reflexly–induced motion of all active EMAs, from their initial stimulus−state totheir final response−state, along the family of extremal (i.e., Euler–Lagrangian) pathsximin(t). (S[x] is properly derived in (8–9) below.)

2 Sub-Cerebellar Biodynamics and Its Spinal Reflex Servo–

Control

Subcerebellar biodynamics includes the following three components: (i) local muscle–jointmechanics, (ii) whole–body musculo–skeletal dynamics, and (iii) autogenetic reflex servo–control.

2.1 Local Muscle–Joint Mechanics

Local muscle–joint mechanics comprises of [Iva06, II06a, II06b]):

2Note that we could also use a reward–based, reinforcement learning rule [SB98], in which system learnsits optimal policy:

innovation(t) = |reward(t)− penalty(t)|.

4

1. Synovial joint dynamics, giving the first stabilizing effect to the conservative skeletondynamics, is described by the (x, x)–form of the Rayleigh – Van der Pol’s dissipationfunction

R =1

2

n∑

i=1

(xi)2 [αi + βi(xi)2],

where αi and βi denote dissipation parameters. Its partial derivatives give rise to theviscous–damping torques and forces in the joints

F jointi = ∂R/∂xi,

which are linear in xi and quadratic in xi.2. Muscular dynamics, giving the driving torques and forces Fmuscle

i = Fmusclei (t, x, x)

with (i = 1, . . . , n) for RHB, describes the internal excitation and contraction dynamicsof equivalent muscular actuators [Hat78].

(a) Excitation dynamics can be described by an impulse force–time relation

F impi = F 0

i (1 − e−t/τi) if stimulation > 0

F impi = F 0

i e−t/τi if stimulation = 0,

where F 0i denote the maximal isometric muscular torques and forces, while τi denote the

associated time characteristics of particular muscular actuators. This relation representsa solution of the Wilkie’s muscular active–state element equation [Wil56]

µ + γ µ = γ S A, µ(0) = 0, 0 < S < 1,

where µ = µ(t) represents the active state of the muscle, γ denotes the element gain,A corresponds to the maximum tension the element can develop, and S = S(r) is the‘desired’ active state as a function of the motor unit stimulus rate r. This is the basis forthe RHB force controller.

(b) Contraction dynamics has classically been described by the Hill’s hyperbolic force–velocity relation [Hil38]

FHilli =

(

F 0i bi − δijaix

j)

(δij xj + bi),

where ai and bi denote the Hill’s parameters, corresponding to the energy dissipated duringthe contraction and the phosphagenic energy conversion rate, respectively, while δij is theKronecker’s δ−tensor.

In this way, RHB describes the excitation/contraction dynamics for the ith equivalentmuscle–joint actuator, using the simple impulse–hyperbolic product relation

Fmusclei (t, x, x) = F imp

i × FHilli .

5

Now, for the purpose of biomedical engineering and rehabilitation, RHB has developedthe so–called hybrid rotational actuator. It includes, along with muscular and viscousforces, the D.C. motor drives, as used in robotics [VBS90, Iva06, II06a]

Frobok = ik(t)− Jkxk(t)−Bkxk(t),

withlkik(t) +Rkik(t) + Ckxk(t) = uk(t),

where k = 1, . . . , n, ik(t) and uk(t) denote currents and voltages in the rotors of thedrives, Rk, lk and Ck are resistances, inductances and capacitances in the rotors, respec-tively, while Jk and Bk correspond to inertia moments and viscous dampings of the drives,respectively.

Finally, to make the model more realistic, we need to add some stochastic torques andforces [IS01, II07a]

Fstochi = Bij[x

i(t), t] dW j(t)

where Bij[x(t), t] represents continuous stochastic diffusion fluctuations, and W j(t) is anN−variable Wiener process (i.e. generalized Brownian motion), with dW j(t) = W j(t +dt)−W j(t) for j = 1, . . . , N .

2.2 Hamiltonian Biodynamics and Its Reflex Servo–Control

General form of Hamiltonian biodynamics on the configuration manifold of human motionis formulated in [IS01, Iva02, Iva04, IB05, II06a, II06c]) using the concept of Euclideangroup of motions SE(3)3 (see Figure 1),

Briefly, based on affine Hamiltonian function of human motion, formally Ha : T ∗Q →R, in local canonical coordinates on the symplectic phase space (which is the cotangent

3Briefly, the Euclidean SE(3)–group is defined as a semidirect (noncommutative) product of 3D rotationsand 3D translations, SE(3) := SO(3)⊲ R

3. Its most important subgroups are the following (for technicaldetails see [II06c, PC05, II07c]):

Subgroup Definition

SO(3), group of rotationsin 3D (a spherical joint)

Set of all proper orthogonal3× 3− rotational matrices

SE(2), special Euclidean groupin 2D (all planar motions)

Set of all 3× 3−matrices:2

4

cos θ sin θ rx− sin θ cos θ ry

0 0 1

3

5

SO(2), group of rotations in 2Dsubgroup of SE(2)–group

(a revolute joint)

Set of all proper orthogonal2× 2− rotational matricesincluded in SE(2)− group

R3, group of translations in 3D(all spatial displacements)

Euclidean 3D vector space

6

Figure 1: The configuration manifold Q of the human musculoskeletal dynamics is definedas an anthropomorphic product of constrained Euclidean SE(3)–groups acting in all major(synovial) human joints.

bundle of the human configuration manifold Q) T ∗Q given as

Ha(x, p, u) = H0(x, p)−Hj(x, p)uj , (5)

where H0(x, p) = Ek(p) + Ep(x) is the physical Hamiltonian (kinetic + potential energy)dependent on joint coordinates xi and their canonical momenta pi, H

j = Hj(x, p), (j =1, . . . , m ≤ n are the coupling Hamiltonians corresponding to the system’s active joints andui = ui(t, x, p) are (reflex) feedback–controls. Using (5) we come to the affine Hamiltoniancontrol RHB–system, in deterministic form

xi = ∂piH0 − ∂piHj uj + ∂piR, (6)

pi = Fi − ∂xiH0 + ∂xiHj uj + ∂xiR,

oi = −∂uiHa = Hj,

xi(0) = xi0, pi(0) = p0i ,

(i = 1, . . . , n; j = 1, . . . , Q ≤ n),

(where ∂u ≡ ∂/∂u, Fi = Fi(t, x, p), H0 = H0(x, p), Hj = Hj(x, p), Ha = Ha(x, p, u),

7

R = R(x, p)), as well as in the fuzzy–stochastic form [IS01, II07a]

dqi =(

∂piH0(σµ)− ∂piHj(σµ)uj + ∂piR

)

dt,

dpi = Bij[xi(t), t] dW j(t) + (7)

(

Fi − ∂xiH0(σµ) + ∂xiHj(σµ)uj + ∂xiR)

dt,

doi = −∂uiHa(σµ) dt = Hj(σµ) dt,

xi(0) = xi0, pi(0) = p0i

In (6)–(7), R = R(x, p) denotes the joint (nonlinear) dissipation function, oi are affinesystem outputs (which can be different from joint coordinates); {σ}µ (with µ ≥ 1) denotefuzzy sets of conservative parameters (segment lengths, masses and moments of inertia),dissipative joint dampings and actuator parameters (amplitudes and frequencies), whilethe bar (.) over a variable denotes the corresponding fuzzified variable; Bij [q

i(t), t] denotediffusion fluctuations and W j(t) are discontinuous jumps as the n–dimensional Wienerprocess.

In this way, the force RHB servo–controller is formulated as affine control Hamiltonian–systems (6–7), which resemble the autogenetic motor servo (see Appendix), acting on thespinal–reflex level of the human locomotion control. A voluntary contraction force F ofhuman skeletal muscle is reflexly excited (positive feedback +F−1) by the responses ofits spindle receptors to stretch and is reflexly inhibited (negative feedback −F−1) by theresponses of its Golgi tendon organs to contraction. Stretch and unloading reflexes aremediated by combined actions of several autogenetic neural pathways, forming the so–called ‘motor servo.’ The term ‘autogenetic’ means that the stimulus excites receptorslocated in the same muscle that is the target of the reflex response. The most importantof these muscle receptors are the primary and secondary endings in the muscle–spindles,which are sensitive to length change – positive length feedback +F−1, and the Golgitendon organs, which are sensitive to contractile force – negative force feedback −F−1.

The gain G of the length feedback +F−1 can be expressed as the positional stiffness(the ratio G ≈ S = dF/dx of the force–F change to the length–x change) of the musclesystem. The greater the stiffness S, the less the muscle will be disturbed by a changein load. The autogenetic circuits +F−1 and −F−1 appear to function as servoregulatoryloops that convey continuously graded amounts of excitation and inhibition to the large(alpha) skeletomotor neurons. Small (gamma) fusimotor neurons innervate the contractilepoles of muscle spindles and function to modulate spindle–receptor discharge.

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3 Cerebellum: The Adaptive Path–Integral Comparator

3.1 Cerebellum as a Neural Controller

Having, thus, defined the spinal reflex control level, we proceed to model the top subcorti-cal commander/controller, the cerebellum (see Appendix). The cerebellum is responsiblefor coordinating precisely timed 〈out|in〉 activity by integrating motor output with ongo-ing sensory feedback (see Figure 2). It receives extensive projections from sensory–motorareas of the cortex and the periphery and directs it back to premotor and motor cortex[Ghe90, Ghe91]. This suggests a role in sensory–motor integration and the timing andexecution of human movements. The cerebellum stores patterns of motor control for fre-quently performed movements, and therefore, its circuits are changed by experience andtraining. It was termed the adjustable pattern generator in the work of J. Houk and col-laborators [HBB96]. Also, it has become the inspiring ‘brain–model’ in robotic research[SA98, Sch98, Sch99].

Figure 2: Schematic 〈out|in〉 organization of the primary cerebellar circuit. In essence,excitatory inputs, conveyed by collateral axons of Mossy and Climbing fibers activate di-rectly neurones in the Deep cerebellar nuclei. The activity of these latter is also modulatedby the inhibitory action of the cerebellar cortex, mediated by the Purkinje cells.

The cerebellum is known to be involved in the production and learning of smoothcoordinated movements [TGK92, FSB97]. Two classes of inputs carry information into the

9

cerebellum: the mossy fibers (MFs) and the climbing fibers (CFs). The MFs provide bothplant state and contextual information [BC81]. The CFs, on the other hand, are thoughtto provide information that reflect errors in recently generated movements [Ito84, Ito90].This information is used to adjust the programs encoded by the cerebellum. The MFscarry plant state, motor efference, and other contextual signals into the cerebellum. Thesefibers impinge on granule cells, whose axons give rise to parallel fibers (PFs). Throughthe combination of inputs from multiple classes of MFs and local inhibitory interneurons,the granule cells are thought to provide a sparse expansive encoding of the incoming stateinformation [Alb71]. The large number of PFs converge on a much smaller set of Purkinjecells (PCs), while the PCs, in turn, provide inhibitory signals to a single cerebellar nuclearcell [FSB97]. Using this principle, the Cerebellar Model Arithmetic Computer, or CMAC–neural network has been built [Alb71, MGK92] and implemented in robotics [Sma98], usingtrial-and-error learning to produce bursts of muscular activity for controlling robot arms.

So, this ‘cerebellar control’ works for simple robotic problems, like non-redundantmanipulation. However, comparing the number of its neurons (107 − 108), to the sizeof conventional neural networks (including CMAC), suggests that artificial neural netscannot satisfactorily model the function of this sophisticated ‘super–bio–computer’, as itsdimensionality is virtually infinite. Despite a lot of research dedicated to its structure andfunction (see [HBB96] and references there cited), the real nature of the cerebellum stillremains a ‘mystery’.

Figure 3: The cerebellum as a motor controller.

The main function of the cerebellum as a motor controller is depicted in Figure 3. Acoordinated movement is easy to recognize, but we know little about how it is achieved.In search of the neural basis of coordination, a model of spinocerebellar interactions was

10

recently presented in [AG05], in which the structural and functional organizing principleis a division of the cerebellum into discrete micro–complexes. Each micro–complex is therecipient of a specific motor error signal, that is, a signal that conveys information about aninappropriate movement. These signals are encoded by spinal reflex circuits and conveyedto the cerebellar cortex through climbing fibre afferents. This organization reveals salientfeatures of cerebellar information processing, but also highlights the importance of systemslevel analysis for a fuller understanding of the neural mechanisms that underlie behavior.

3.2 Hamiltonian Action and Neural Path Integral

Here, we propose a quantum–like adaptive control approach to modeling the ‘cerebellarmystery’. Corresponding to the affine Hamiltonian control function (5) we define the affine

Hamiltonian control action,

Saff [q, p] =

∫ tout

tin

dτ[

piqi −Haff (q, p)

]

. (8)

From the affine Hamiltonian action (8) we further derive the associated expression forthe neural phase–space path integral (in normal units), representing the cerebellar sensory–motor amplitude 〈out|in〉,

⟨

qiout, pouti |qiin, p

ini

⟩

=

∫

D[w, q, p] ei Saff [q,p] (9)

=

∫

D[w, q, p] exp

{

i

∫ tout

tin

dτ[

piqi −Haff (q, p)

]

}

,

with

∫

D[w, q, p] =

∫ n∏

τ=1

wi(τ)dpi(τ)dqi(τ)

2π,

where wi = wi(t) denote the cerebellar synaptic weights positioned along its neural path-ways, being continuously updated using the Hebbian–like self–organizing learning rule(3). Given the transition amplitude out|in (9), the cerebellar sensory–motor transition

probability is defined as its absolute square, |〈out|in〉|2.In the phase–space path integral (9), qiin = qiin(t), qiout = qiout(t); pini = pini (t), pouti =

pouti (t); tin ≤ t ≤ tout, for all discrete time steps, t = 1, ..., n → ∞, and we are allowingfor the affine Hamiltonian Haff (q, p) to depend upon all the (M ≤ N) EMA–anglesand angular momenta collectively. Here, we actually systematically took a discretizeddifferential time limit of the form tσ− tσ−1 ≡ dτ (both σ and τ denote discrete time steps)

and wrote(qiσ−qiσ−1)

(tσ−tσ−1)≡ qi. For technical details regarding the path integral calculations on

Riemannian and symplectic manifolds (including the standard regularization procedures),see [Kla97, Kla00].

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Now, motor learning occurring in the cerebellum can be observed using functional MRimaging, showing changes in the cerebellar action potential, related to the motor tasks(see, e.g., [MA02]). To account for these electro–physiological currents, we need to addthe source term Ji(t)q

i(t) to the affine Hamiltonian action (8), (the current Ji = Ji(t) actsas a source JiA

i of the cerebellar electrical potential Ai = Ai(t)),

Saff [q, p, J ] =

∫ tout

tin

dτ[

piqi −Haff (q, p) + Jiq

i]

,

which, subsequently gives the cerebellar path integral with the action potential source,coming either from the motor cortex or from other subcortical areas.

Note that the standard Wick rotation: t 7→ t (see [Kla97, Kla00]), makes our pathintegral real, i.e.,

∫

D[w, q, p] ei Saff [q,p] Wick−−−→

∫

D[w, q, p] e− Saff [q,p],

while their subsequent discretization gives the standard thermodynamic partition function

(see Appendix),

Z =∑

j

−wjEj/T , (10)

where Ej is the energy eigenvalue corresponding to the affine Hamiltonian Haff (q, p), T isthe temperature–like environmental control parameter, and the sum runs over all energyeigenstates (labelled by the index j). From (10), we can further calculate all statisticaland thermodynamic system properties (see [Fey72]), as for example, transition entropy

S = kB lnZ, etc.

3.3 Entropy and Motor Control

Our cerebellar path integral controller is closely related to entropic motor control [HN08a,HN08b], which deals with neuro-physiological feedback information and environmentaluncertainty. The probabilistic nature of human motor action can be characterized by en-tropies at the level of the organism, task, and environment. Systematic changes in motoradaptation are characterized as task–organism and environment–organism tradeoffs in en-tropy. Such compensatory adaptations lead to a view of goal–directed motor control as theproduct of an underlying conservation of entropy across the task–organism–environmentsystem. In particular, an experiment conducted in [HN08b] examined the changes in en-tropy of the coordination of isometric force output under different levels of task demandsand feedback from the environment. The goal of the study was to examine the hypothesisthat human motor adaptation can be characterized as a process of entropy conservationthat is reflected in the compensation of entropy between the task, organism motor output,

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and environment. Information entropy of the coordination dynamics relative phase of themotor output was made conditional on the idealized situation of human movement, forwhich the goal was always achieved. Conditional entropy of the motor output decreasedas the error tolerance and feedback frequency were decreased. Thus, as the likelihood ofmeeting the task demands was decreased increased task entropy and/or the amount of in-formation from the environment is reduced increased environmental entropy, the subjectsof this experiment employed fewer coordination patterns in the force output to achieve thegoal. The conservation of entropy supports the view that context dependent adaptationsin human goal–directed action are guided fundamentally by natural law and provides anovel means of examining human motor behavior. This is fundamentally related to theHeisenberg uncertainty principle [II08b] and further supports the argument for the primacyof a probabilistic approach toward the study of biodynamic cognition systems.

The action–amplitude formalism represents a kind of a generalization of the Haken-Kelso-Bunz (HKB) model of self-organization in the individual’s motor system [HKB85,Kel95], including: multi-stability, phase transitions and hysteresis effects, presenting acontrary view to the purely feedback driven systems. HKB uses the concepts of synergetics(order parameters, control parameters, instability, etc) and the mathematical tools ofnonlinearly coupled (nonlinear) dynamical systems to account for self-organized behaviorboth at the cooperative, coordinative level and at the level of the individual coordinatingelements. The HKB model stands as a building block upon which numerous extensionsand elaborations have been constructed. In particular, it has been possible to derive itfrom a realistic model of the cortical sheet in which neural areas undergo a reorganizationthat is mediated by intra- and inter-cortical connections. Also, the HKB model describesphase transitions (‘switches’) in coordinated human movement as follows: (i) when theagent begins in the anti-phase mode and speed of movement is increased, a spontaneousswitch to symmetrical, in-phase movement occurs; (ii) this transition happens swiftly ata certain critical frequency; (iii) after the switch has occurred and the movement rate isnow decreased the subject remains in the symmetrical mode, i.e. she does not switchback; and (iv) no such transitions occur if the subject begins with symmetrical, in-phasemovements. The HKB dynamics of the order parameter relative phase as is given by anonlinear first-order ODE:

φ = (α+ 2βr2) sin φ− βr2 sin 2φ,

where φ is the phase relation (that characterizes the observed patterns of behavior, changesabruptly at the transition and is only weakly dependent on parameters outside the phasetransition), r is the oscillator amplitude, while α, β are coupling parameters (from whichthe critical frequency where the phase transition occurs can be calculated).

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4 Appendix

4.1 Houk’s Autogenetic Motor Servo

About three decades ago, James Houk pointed out in [Hou67, HSG70, Hou78, Hou79] thatstretch and unloading reflexes were mediated by combined actions of several autogeneticneural pathways. In this context, “autogenetic” (or, autogenic) means that the stimulusexcites receptors located in the same muscle that is the target of the reflex response.The most important of these muscle receptors are the primary and secondary endingsin muscle spindles, sensitive to length change, and the Golgi tendon organs, sensitive tocontractile force. The autogenetic circuits appear to function as servo-regulatory loopsthat convey continuously graded amounts of excitation and inhibition to the large (alpha)skeletomotor neurons. Small (gamma) fusimotor neurons innervate the contractile poles ofmuscle spindles and function to modulate spindle–receptor discharge. Houk’s term “motorservo” [Hou78] has been used to refer to this entire control system, summarized by theblock diagram in Figure 4.

Figure 4: Houk’s autogenetic motor servo.

Prior to a study by Matthews [Mat69], it was widely assumed that secondary endings

14

belong to the mixed population of “flexor reflex afferents,” so called because their activa-tion provokes the flexor reflex pattern – excitation of flexor motoneurons and inhibition ofextensor motoneurons. Matthews’ results indicated that some category of muscle stretchreceptor other than the primary ending provides important excitation to extensor muscles,and he argued forcefully that it must be the secondary ending.

The primary and secondary muscle spindle afferent fibers both arise from a specializedstructure within the muscle, the muscle spindle, a fusiform structure 4–7 mm long and 80–200 µ in diameter. The spindles are located deep within the muscle mass, scattered widelythrough the muscle body, and attached to the tendon, the endomysium or the perimysium,so as to be in parallel with the extrafusal or regular muscle fibers. Although spindles arescattered widely in muscles, they are not found throughout. Muscle spindle (see Figure??) contains two types of intrafusal muscle fibers (intrafusal means inside the fusiformspindle): the nuclear bag fibers and the nuclear chain fibers. The nuclear bag fibers arethicker and longer than the nuclear chain fibers, and they receive their name from theaccumulation of their nuclei in the expanded bag-like equatorial region-the nuclear bag.The nuclear chain fibers have no equatorial bulge; rather their nuclei are lined up in theequatorial region-the nuclear chain. A typical spindle contains two nuclear bag fibers and4-5 nuclear chain fibers.

The pathways from primary and secondary endings are treated commonly by Houkin Figure 4, since both receptors are sensitive to muscle length and both provoke reflexexcitation. However, primary endings show an additional sensitivity to the dynamic phaseof length change, called dynamic responsiveness, and they also show a much–enhancedsensitivity to small changes in muscle length [Mat72].

The motor servo comprises three closed circuits (Figure 4), two neural feedback path-ways, and one circuit representing the mechanical interaction between a muscle and itsload. One of the feedback pathways, that from spindle receptors, conveys informationconcerning muscle length, and it follows that this loop will act to keep muscle lengthconstant. The other feedback pathway, that from tendon organs, conveys informationconcerning muscle force, and it acts to keep force constant.

In general, it is physically impossible to maintain both muscle length and force con-stant when external loads vary; in this situation the action of the two feedback loops willoppose each other. For example, an increased load force will lengthen the muscle and causemuscular force to increase as the muscle is stretched out on its length-tension curve. Theincreased length will lead to excitation of motoneurons, whereas the increased force willlead to inhibition. It follows that the net regulatory action conveyed by skeletomotor out-put will depend on some relationship between force change and length change and on thestrength of the feedback from muscle spindles and tendon organs. A simple mathematicalderivation [NH76] demonstrates that the change in skeletomotor output, the error signalof the motor servo, Should be proportional to the difference between a regulated stiffnessand the actual stiffness provided by the mechanical properties of the muscle, where stiff-

15

ness has the units of force change divided by length change. The regulated stiffness isdetermined by the ratio of the gain of length to force feedback.

It follows that the combination of spindle receptor and tendon organ feedback willtend to maintain the stiffness of the neuromuscular apparatus at some regulated level. Ifthis level is high, due to a high gain of length feedback and a low gain of force feedback,one could simply forget about force feedback and treat muscle length as the regulatedvariable of the system. However, if the regulated level of stiffness is intermediate in value,i.e. not appreciably different from the average stiffness arising from muscle mechanicalproperties in the absence of reflex actions, one would conclude that stiffness, or its inverse,compliance, is the regulated property of the motor servo.

In this way, the autogenetic reflex motor servo provides the local, reflex feedbackloops for individual muscular contractions. A voluntary contraction force F of humanskeletal muscle is reflexly excited (positive feedback +F−1) by the responses of its spindlereceptors to stretch and is reflexly inhibited (negative feedback −F−1) by the responsesof its Golgi tendon organs to contraction. Stretch and unloading reflexes are mediated bycombined actions of several autogenetic neural pathways, forming the motor servo (see[II06a, II06b, II06c]).

In other words, branches of the afferent fibers also synapse with with interneuronsthat inhibit motor neurons controlling the antagonistic muscles – reciprocal inhibition.Consequently, the stretch stimulus causes the antagonists to relax so that they cannotresists the shortening of the stretched muscle caused by the main reflex arc. Similarly,firing of the Golgi tendon receptors causes inhibition of the muscle contracting too strongand simultaneous reciprocal activation of its antagonist.

4.2 Cerebellum and Muscular Synergy

The cerebellum is a brain region anatomically located at the bottom rear of the head(the hindbrain), directly above the brainstem, which is important for a number of subcon-scious and automatic motor functions, including motor learning. It processes informationreceived from the motor cortex, as well as from proprioceptors and visual and equilibriumpathways, and gives ‘instructions’ to the motor cortex and other subcortical motor centers(like the basal nuclei), which result in proper balance and posture, as well as smooth, co-ordinated skeletal movements, like walking, running, jumping, driving, typing, playing thepiano, etc. Patients with cerebellar dysfunction have problems with precise movements,such as walking and balance, and hand and arm movements. The cerebellum looks simi-

lar in all animals, from fish to mice to humans. This has been taken as evidence that itperforms a common function, such as regulating motor learning and the timing of move-ments, in all animals. Studies of simple forms of motor learning in the vestibulo–ocularreflex and eye–blink conditioning are demonstrating that timing and amplitude of learnedmovements are encoded by the cerebellum.

16

When someone compares learning a new skill to learning how to ride a bike they implythat once mastered, the task seems imbedded in our brain forever. Well, imbedded in thecerebellum to be exact. This brain structure is the commander of coordinated movementand possibly even some forms of cognitive learning. Damage to this area leads to motoror movement difficulties.

A part of a human brain that is devoted to the sensory-motor control of human move-ment, that is motor coordination and learning, as well as equilibrium and posture, is thecerebellum (which in Latin means “little brain”). It performs integration of sensory per-ception and motor output. Many neural pathways link the cerebellum with the motorcortex, which sends information to the muscles causing them to move, and the spino–cerebellar tract, which provides proprioception, or feedback on the position of the bodyin space. The cerebellum integrates these pathways, using the constant feedback on bodyposition to fine–tune motor movements [Ito84].

The human cerebellum has 7–14 million Purkinje cells. Each receives about 200,000synapses, most onto dendritic splines. Granule cell axons form the parallel fibers. Theymake excitatory synapses onto Purkinje cell dendrites. Each parallel fibre synapses onabout 200 Purkinje cells. They create a strip of excitation along the cerebellar folia.

Mossy fibers are one of two main sources of input to the cerebellar cortex (see Figure5). A mossy fibre is an axon terminal that ends in a large, bulbous swelling. These mossyfibers enter the granule cell layer and synapse on the dendrites of granule cells; in fact thegranule cells reach out with little ‘claws’ to grasp the terminals. The granule cells thensend their axons up to the molecular layer, where they end in a T and run parallel to thesurface. For this reason these axons are called parallel fibers. The parallel fibers synapseon the huge dendritic arrays of the Purkinje cells. However, the individual parallel fibersare not a strong drive to the Purkinje cells. The Purkinje cell dendrites fan out within aplane, like the splayed fingers of one hand. If we were to turn a Purkinje cell to the side, itwould have almost no width at all. The parallel fibers run perpendicular to the Purkinjecells, so that they only make contact once as they pass through the dendrites.

Unless firing in bursts, parallel fibre EPSPs do not fire Purkinje cells. Parallel fibersprovide excitation to all of the Purkinje cells they encounter. Thus, granule cell activityresults in a strip of activated Purkinje cells.

Mossy fibers arise from the spinal cord and brainstem. They synapse onto granule cellsand deep cerebellar nuclei. The Purkinje cell makes an inhibitory synapse (GABA) to thedeep nuclei. Mossy fibre input goes to both cerebellar cortex and deep nuclei. When thePurkinje cell fires, it inhibits output from the deep nuclei.

The climbing fibre arises from the inferior olive. It makes about 300 excitatory synapsesonto one Purkinje cell. This powerful input can fire the Purkinje cell.

The parallel fibre synapses are plastic—that is, they can be modified by experience.When parallel fibre activity and climbing fibre activity converge on the same Purkinje cell,the parallel fibre synapses become weaker (EPSPs are smaller). This is called long-term

17

Figure 5: Stereotypical ways throughout the cerebellum.

depression. Weakened parallel fibre synapses result in less Purkinje cell activity and lessinhibition to the deep nuclei, resulting in facilitated deep nuclei output. Consequently,the mossy fibre collaterals control the deep nuclei.

The basket cell is activated by parallel fibers afferents. It makes inhibitory synapsesonto Purkinje cells. It provides lateral inhibition to Purkinje cells. Basket cells inhibitPurkinje cells lateral to the active beam.

Golgi cells receive input from parallel fibers, mossy fibers, and climbing fibers. Theyinhibit granule cells. Golgi cells provide feedback inhibition to granule cells as well asfeedforward inhibition to granule cells. Golgi cells create a brief burst of granule cellactivity.

Although each parallel fibre touches each Purkinje cell only once, the thousands ofparallel fibers working together can drive the Purkinje cells to fire like mad.

The second main type of input to the folium is the climbing fibre. The climbing fibersgo straight to the Purkinje cell layer and snake up the Purkinje dendrites, like ivy climbinga trellis. Each climbing fibre associates with only one Purkinje cell, but when the climbingfibre fires, it provokes a large response in the Purkinje cell.

The Purkinje cell compares and processes the varying inputs it gets, and finally sendsits own axons out through the white matter and down to the deep nuclei. Although theinhibitory Purkinje cells are the main output of the cerebellar cortex, the output from the

18

cerebellum as a whole comes from the deep nuclei. The three deep nuclei are responsiblefor sending excitatory output back to the thalamus, as well as to postural and vestibularcenters.

There are a few other cell types in cerebellar cortex, which can all be lumped into thecategory of inhibitory interneuron. The Golgi cell is found among the granule cells. Thestellate and basket cells live in the molecular layer. The basket cell (right) drops axonbranches down into the Purkinje cell layer where the branches wrap around the cell bodieslike baskets.

The cerebellum operates in 3’s: there are 3 highways leading in and out of the cere-bellum, there are 3 main inputs, and there are 3 main outputs from 3 deep nuclei. Theyare:

The 3 highways are the peduncles. There are 3 pairs (see [Mol97, Har97, Mar98]):

1. The inferior cerebellar peduncle (restiform body) contains the dorsal spinocerebellartract (DSCT) fibers. These fibers arise from cells in the ipsilateral Clarke’s col-umn in the spinal cord (C8–L3). This peduncle contains the cuneo–cerebellar tract(CCT) fibers. These fibers arise from the ipsilateral accessory cuneate nucleus. Thelargest component of the inferior cerebellar peduncle consists of the olivo–cerebellartract (OCT) fibers. These fibers arise from the contralateral inferior olive. Finally,vestibulo–cerebellar tract (VCT) fibers arise from cells in both the vestibular gan-glion and the vestibular nuclei and pass in the inferior cerebellar peduncle to reachthe cerebellum.

2. The middle cerebellar peduncle (brachium pontis) contains the pontocerebellar tract(PCT) fibers. These fibers arise from the contralateral pontine grey.

3. The superior cerebellar peduncle (brachium conjunctivum) is the primary efferent(out of the cerebellum) peduncle of the cerebellum. It contains fibers that arisefrom several deep cerebellar nuclei. These fibers pass ipsilaterally for a while andthen cross at the level of the inferior colliculus to form the decussation of the superiorcerebellar peduncle. These fibers then continue ipsilaterally to terminate in the rednucleus (‘ruber–duber’) and the motor nuclei of the thalamus (VA, VL).

The 3 inputs are: mossy fibers from the spinocerebellar pathways, climbing fibers fromthe inferior olive, and more mossy fibers from the pons, which are carrying informationfrom cerebral cortex (see Figure 6). The mossy fibers from the spinal cord have come upipsilaterally, so they do not need to cross. The fibers coming down from cerebral cortex,however, do need to cross (as the cerebrum is concerned with the opposite side of thebody, unlike the cerebellum). These fibers synapse in the pons (hence the huge block offibers in the cerebral peduncles labelled ‘cortico–pontine’), cross, and enter the cerebellumas mossy fibers.

19

Figure 6: Inputs and outputs of the cerebellum.

The 3 deep nuclei are the fastigial, interposed, and dentate nuclei. The fastigial nu-cleus is primarily concerned with balance, and sends information mainly to vestibular andreticular nuclei. The dentate and interposed nuclei are concerned more with voluntarymovement, and send axons mainly to thalamus and the red nucleus.

The main function of the cerebellum as a motor controller is depicted in Figure 3. Acoordinated movement is easy to recognize, but we know little about how it is achieved.In search of the neural basis of coordination, a model of spinocerebellar interactions wasrecently presented in [AG05], in which the structure-functional organizing principle isa division of the cerebellum into discrete micro–complexes. Each micro–complex is therecipient of a specific motor error signal, that is, a signal that conveys information about aninappropriate movement. These signals are encoded by spinal reflex circuits and conveyedto the cerebellar cortex through climbing fibre afferents. This organization reveals salientfeatures of cerebellar information processing, but also highlights the importance of systemslevel analysis for a fuller understanding of the neural mechanisms that underlie behavior.

The authors of [AG05] reviewed anatomical and physiological foundations of cerebellarinformation processing. The cerebellum is crucial for the coordination of movement. Theauthors presented a model of the cerebellar paravermis, a region concerned with the controlof voluntary limb movements through its interconnections with the spinal cord. Theyparticularly focused on the olivo-cerebellar climbing fibre system.

20

Climbing fibres are proposed to convey motor error signals (signals that convey in-formation about inappropriate movements) related to elementary limb movements thatresult from the contraction of single muscles. The actual encoding of motor error signalsis suggested to depend on sensorimotor transformations carried out by spinal modules thatmediate nociceptive withdrawal reflexes.

The termination of the climbing fibre system in the cerebellar cortex subdivides theparavermis into distinct microzones. Functionally similar but spatially separate micro-zones converge onto a common group of cerebellar nuclear neurons. The processing unitsformed as a consequence are termed ‘multizonal micro-complexes’ (MZMCs), and are eachrelated to a specific spinal reflex module.

The distributed nature of microzones that belong to a given MZMC is proposed toenable similar climbing fibre inputs to integrate with mossy fibre inputs that arise fromdifferent sources. Anatomical results consistent with this notion have been obtained.

Within an individual MZMC, the skin receptive fields of climbing fibres, mossy fibresand cerebellar cortical inhibitory interneurons appear to be similar. This indicates thatthe inhibitory receptive fields of Purkinje cells within a particular MZMC result from theactivation of inhibitory interneurons by local granule cells.

On the other hand, the parallel fibre–mediated excitatory receptive fields of the Purk-inje cells in the same MZMC differ from all of the other receptive fields, but are similarto those of mossy fibres in another MZMC. This indicates that the excitatory input toPurkinje cells in a given MZMC originates in non–local granule cells and is mediated oversome distance by parallel fibres.

The output from individual MZMCs often involves two or three segments of the ipsi-lateral limb, indicative of control of multi–joint muscle synergies. The distal–most musclein this synergy seems to have a roughly antagonistic action to the muscle associated withthe climbing fibre input to the MZMC.

The model proposed in [AG05] indicates that the cerebellar paravermis system couldprovide the control of both single– and multi–joint movements. Agonist-antagonist activityassociated with single–joint movements might be controlled within a particular MZMC,whereas coordination across multiple joints might be governed by interactions betweenMZMCs, mediated by parallel fibres.

Two main theories address the function of the cerebellum, both dealing with motorcoordination. One claims that the cerebellum functions as a regulator of the “timingof movements.” This has emerged from studies of patients whose timed movements aredisrupted [IKD88].

The second, “Tensor Network Theory” provides a mathematical model of transforma-tion of sensory (covariant) space-time coordinates into motor (contravariant) coordinatesby cerebellar neuronal networks [PL80, PL82, PL85].

Studies of motor learning in the vestibulo–ocular reflex and eye-blink conditioningdemonstrate that the timing and amplitude of learned movements are encoded by the

21

cerebellum [BK04]. Many synaptic plasticity mechanisms have been found throughout thecerebellum. The Marr–Albus model mostly attributes motor learning to a single plasticitymechanism: the long-term depression of parallel fiber synapses. The Tensor NetworkTheory of sensory–motor transformations by the cerebellum has also been experimentallysupported [GZ86].

4.3 Feynman’s Partition Function

Recall that in statistical mechanics, the so–called partition function Z is a quantity thatencodes the statistical properties of a system in thermodynamic equilibrium. It is a func-tion of temperature and other parameters, such as the volume enclosing a gas. Otherthermodynamic variables of the system, such as the total energy, free energy, entropy, andpressure, can be expressed in terms of the partition function or its derivatives.4

The partition function of a canonical ensemble5 is defined as a sum Z(β) =∑

j e−βEj , where β = 1/(kBT ) is the ‘inverse temperature’, where T is an ordinary tem-

perature and kB is the Boltzmann’s constant. However, as the position xi and momentumpi variables of an ith particle in a system can vary continuously, the set of microstates is ac-tually uncountable. In this case, some form of coarse–graining procedure must be carriedout, which essentially amounts to treating two mechanical states as the same microstate ifthe differences in their position and momentum variables are ‘small enough’. The partitionfunction then takes the form of an integral. For instance, the partition function of a gasconsisting of N molecules is proportional to the 6N−dimensional phase–space integral,

Z(β) ∼

∫

R6N

d3pi d3xi exp[−βH(pi, xi)],

where H = H(pi, xi), (i = 1, ..., N) is the classical Hamiltonian (total energy) function.

4There are actually several different types of partition functions, each corresponding to different typesof statistical ensemble (or, equivalently, different types of free energy.) The canonical partition functionapplies to a canonical ensemble, in which the system is allowed to exchange heat with the environment atfixed temperature, volume, and number of particles. The grand canonical partition function applies to agrand canonical ensemble, in which the system can exchange both heat and particles with the environment,at fixed temperature, volume, and chemical potential. Other types of partition functions can be definedfor different circumstances.

5A canonical ensemble is a statistical ensemble representing a probability distribution of microscopicstates of the system. Its probability distribution is characterized by the proportion pi of members of theensemble which exhibit a measurable macroscopic state i, where the proportion of microscopic states foreach macroscopic state i is given by the Boltzmann distribution,

pi =1Ze−Ei/(kT ) = e−(Ei−A)/(kT )

,

where Ei is the energy of state i. It can be shown that this is the distribution which is most likely, ifeach system in the ensemble can exchange energy with a heat bath, or alternatively with a large numberof similar systems. In other words, it is the distribution which has maximum entropy for a given averageenergy < Ei >.

22

More generally, the so–called configuration integral, as used in probability theory, in-formation science and dynamical systems, is an abstraction of the above definition of apartition function in statistical mechanics. It is a special case of a normalizing constant inprobability theory, for the Boltzmann distribution. The partition function occurs in manyproblems of probability theory because, in situations where there is a natural symmetry,its associated probability measure, the Gibbs measure (see below), which generalizes thenotion of the canonical ensemble, has the Markov property.

Given a set of random variables Xi taking on values xi, and purely potential Hamilto-nian function H(xi), (i = 1, ..., N), the partition function is defined as

Z(β) =∑

xi

exp [−βH(xi)] .

The functionH is understood to be a real-valued function on the space of states {X1,X2, · · · }while β is a real-valued free parameter (conventionally, the inverse temperature). The sumover the xi is understood to be a sum over all possible values that the random variableXi may take. Thus, the sum is to be replaced by an integral when the Xi are continuous,rather than discrete. Thus, one writes

Z(β) =

∫

dxi exp [−βH(xi)] ,

for the case of continuously-varying random variables Xi.The Gibbs measure of a random variable Xi having the value xi is defined as the

probability density function

P (Xi = xi) =1

Z(β)exp [−βE(xi)] =

exp [−βH(xi)]∑

xiexp [−βH(xi)]

.

where E(xi) = H(xi) is the energy of the configuration xi. This probability, which isnow properly normalized so that 0 ≤ P (xi) ≤ 1, can be interpreted as a likelihood that aspecific configuration of values xi, (i = 1, 2, ...N) occurs in the system.

As such, the partition function Z(β) can be understood to provide the Gibbs measureon the space of states, which is the unique statistical distribution that maximizes theentropy for a fixed expectation value of the energy,

〈H〉 = −∂ log(Z(β))

∂β.

The associated entropy is given by

S = −∑

xi

P (xi) lnP (xi) = β〈H〉+ logZ(β).

23

The principle of maximum entropy related to the expectation value of the energy 〈H〉,is a postulate about a universal feature of any probability assignment on a given set ofpropositions (events, hypotheses, indices, etc.). Let some testable information about aprobability distribution function be given. Consider the set of all trial probability distri-butions which encode this information. Then the probability distribution which maximizesthe information entropy is the true probability distribution, with respect to the testableinformation prescribed.

Now, the number of variables Xi need not be countable, in which case the set ofcoordinates {xi} becomes a field φ = φ(x), so the sum is to be replaced by the Euclidean

path integral (that is a Wick–rotated Feynman transition amplitude in imaginary time),as

Z(φ) =

∫

D[φ] exp [−H(φ)] .

More generally, in quantum field theory, instead of the field Hamiltonian H(φ) we havethe action S(φ) of the theory. Both Euclidean path integral,

Z(φ) =

∫

D[φ] exp [−S(φ)] , real path integral in imaginary time (11)

and Lorentzian one,

Z(φ) =

∫

D[φ] exp [iS(φ)] , complex path integral in real time, (12)

are usually called ‘partition functions’. While the Lorentzian path integral (12) repre-sents a quantum-field theory-generalization of the Schrodinger equation, the Euclideanpath integral (11) represents a statistical-field-theory generalization of the Fokker–Planckequation.

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